Research on Intelligent Low-Dimensional Manifold Algorithm Based on Heegaard Mathematical Model

The theory of three-dimensional manifold combination topology is an important branch of low-dimensional topology. Heegaard decomposition is one of the important methods in the topology of three-dimensional manifolds. Casson and Gordon introduced the idea of weakly reducible Heegaard decomposition in 1987, which greatly promoted the research process of Heegaard decomposition theory in combinatorial topology. In 2000, Hempel introduced the idea of Heegaard's decomposition distance, which not only expanded Casson-Gordon's theory, but also made the research of Heegaard's decomposition theory precedentially active and made many major breakthroughs in the research of many very difficult problems. This paper mainly discusses the fusion of Heegaard decomposition of three-dimensional manifolds. The main results give some sufficient conditions for the deficiency non-degradation of three-dimensional manifolds after fusion in some cases. Let Ni be a compact, orientable connected three-dimensional manifold, and Gi be an incompressible boundary branch of Ni, f(Gi)
$\geq 1$
and
$\mathrm{G}1\sim \mathrm{G}2. \phi$
:
$\mathrm{G}1\sim \mathrm{G}2$
is a homeomorphism,
$\mathrm{N}=\mathrm{N}1\mathrm{U}\phi \mathrm{N}2, \mathrm{G}=\mathrm{G}2=\phi(\mathrm{G}1)$
. Then (1) If Ni has a Heegaard decomposition Ui U SiWi that satisfies
$\mathrm{Z}(\text{Si})\geq 2\mathrm{f}(\text{Ni})-1, \mathrm{i}=1$
, then
$\mathrm{f}(\mathrm{N})=\mathrm{f}(\mathrm{N}1)+\mathrm{f}(\mathrm{N}2)-\mathrm{f}(\mathrm{G})$
. (2) If Ni has a Heegaard decomposition Ui U SiWi that satisfies Z(Si)
$\geq 2\mathrm{f}(\text{Ni})+1, \mathrm{i}=1,2$
. Then the Heegaard decomposition of the smallest genus of
$\mathrm{f}(\mathrm{N})=\mathrm{f}(\mathrm{N}1)+\mathrm{f}(\mathrm{N}2)-\mathrm{f}(\mathrm{G})$
and N is unique in the sense of a conjunction trace. Let Ki be a knot in the closed three-dimensional manifold Ni, (Ni, Ki)
$(\mathrm{S}2\times \mathrm{S}1, \mathrm{x}0\times \mathrm{S}1)$
, where S2 is a two-dimensional sphere,
$\mathrm{x}0\in$
S2,
$\mathrm{i}=1,2. (\mathrm{N}, \mathrm{K})$
= (N1#N2, K1#K2). If the essential surface Gi of any latitude of the knot Ki satisfies
$\chi(\text{Gi})\leq 1-2\mathrm{t}(\text{Ki}), \mathrm{i}=1,2$
. Then
$\mathrm{t}(\mathrm{K})\geq \mathrm{t}(\mathrm{K}1)+\mathrm{t}$
(K2). Further, under the above assumptions, if the distance of the Heegaard decomposition of any minimal deficiency of E(Ki) and E (K2) is greater than or equal to 3, then
$\mathrm{t}(\mathrm{K})=\mathrm{t}(\mathrm{K}1)+\mathrm{t}(\mathrm{K}2)+1$
. 3) Let Ki be a knot in the closed three-dimensional manifold Ni, where
$\mathrm{i}=1,2$
and
$(\mathrm{N}, \mathrm{K})=(\mathrm{N}1 \#\mathrm{N}2, \mathrm{K}1 \#\mathrm{K}2)$
. Then (1) if the complement E(Ki) of the knot has a Heegaard decomposition Ui
$\cup$
(Si)Wi satisfying Z(Si)
$\geq 2\mathrm{t}(\text{Ki})+1, \mathrm{i}=1,2$
, then
$\mathrm{t}(\mathrm{K})\geq \mathrm{t}(\mathrm{K})1+\mathrm{t}$
(K2). (2) If the complement E(Ki) of the knot has a Heegaard decomposition Ui
$\cup$
(Si)Wi satisfying Z(Si)
$\geq 2\mathrm{t}(\text{Ki})+3, \mathrm{i}=1,2$
, then
$\mathrm{t}(\mathrm{K})=\mathrm{t}(\mathrm{K})1+\mathrm{t}(\mathrm{K}) 2+1$
. Finally, the additivity of the relative genus of a three-dimensional manifold under the connected sum operation is proved.